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Sep 22, 2013
09/13

by
Pierre Arnoux; Thomas A. Schmidt

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We show that each of Veech's original examples of translation surfaces with ``optimal dynamics'' whose trace field is of degree greater than two has non-periodic directions of vanishing SAF-invariant. Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant.

Source: http://arxiv.org/abs/0906.4571v1

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Sep 18, 2013
09/13

by
Thomas A. Schmidt; Mark Sheingorn

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We show that the set of real numbers of Lagrange value 3 has Hausdorff dimension zero by showing the appropriate generalization for each element of the Teichmueller space of the appropriate subgroup of the classical modular group.

Source: http://arxiv.org/abs/1201.4161v1

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Sep 22, 2013
09/13

by
Kariane Calta; Thomas A. Schmidt

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We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of non-parabolic elements in the periodic field of certain translation surfaces.

Source: http://arxiv.org/abs/1210.1293v1

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Sep 19, 2013
09/13

by
Pascal Hubert; Thomas A. Schmidt

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We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by...

Source: http://arxiv.org/abs/1010.3475v1

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12

Sep 20, 2013
09/13

by
Pascal Hubert; Thomas A. Schmidt

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Veech groups uniformize Teichm\"uller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end is the limit of directions of inequivalent infinite ends.

Source: http://arxiv.org/abs/math/0410132v1

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Sep 20, 2013
09/13

by
Thomas A. Schmidt; Mark Sheingorn

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We introduce a new method to establish McShane's Identity, based upon the fact that elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics.

Source: http://arxiv.org/abs/0802.3096v1

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Sep 20, 2013
09/13

by
Pierre Arnoux; Thomas A. Schmidt

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We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced \alpha-variants.

Source: http://arxiv.org/abs/1207.7299v1

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Sep 21, 2013
09/13

by
Cor Kraaikamp; Thomas A. Schmidt; Ionica Smeets

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We give natural extensions for the alpha-Rosen continued fractions of Dajani et al. for a set of small alpha values by appropriately adding and deleting rectangles from the region of the natural extension for the standard Rosen fractions. It follows that the underlying maps have equal entropy.

Source: http://arxiv.org/abs/0905.4588v1

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Sep 22, 2013
09/13

by
Cor Kraaikamp; Hitoshi Nakada; Thomas A. Schmidt

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A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.

Source: http://arxiv.org/abs/0812.0548v1

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Jul 20, 2013
07/13

by
Yann Bugeaud; Pascal Hubert; Thomas A. Schmidt

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We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

Source: http://arxiv.org/abs/1007.2050v1

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Sep 19, 2013
09/13

by
Cor Kraaikamp; Thomas A. Schmidt; Ionica Smeets

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The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.

Source: http://arxiv.org/abs/0708.3257v1

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Sep 21, 2013
09/13

by
Cor Kraaikamp; Thomas A. Schmidt; Wolfgang Steiner

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We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le 1$, the product of the entropy with the measure of the domain equals $\pi^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$.

Source: http://arxiv.org/abs/1011.4283v3